Linear Algebra Proof

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avakm55

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Hi! I am not sure where to start on this proof and would love some help:)
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I do know the definition of spanning. Spanning means that a certain sub space can be written as a linear combination (so w=c1v1+c2v2+...+cnvn) of a vector space. What I don’t understand is how to go about setting this up.

The thing I tried was writing out a matrix with all of the vectors, multiplied by the constant matrix, and setting it equal to w, but this didn't help me with anything.

I feel like I have all of these definitions but am unsure about how to apply them.
 
Clearly vk is not in T. If T spans V, then vk would be a linear combination of the vectors in T. That is vk = c1v1 + c2v2 + ... + ck-1vk-1. Continue from here.
 
Is there a rule that I can find somewhere that says if vector set A does not span vector set B, then the vector space that A forms a basis for cannot be the vector space that B forms a basis for?
 
I think I have had a relevation, but can someone correct me if I'm wrong. Since vk is an element of S, and because S spans V, T cannot span V because it does not span S. And since every vector in V can be written as a linear combination of S, every vector of V can not be written as a linear combination of T.

Is explaining this enough to justify my conclusion? I can't think of a more numerical way to show this.
 
I will give you this one but from here on you need to find the solutions/proofs with our help.

This is a continuation of my previous post.

vk = c1v1 + c2v2 + ... + ck-1vk-1

Then vk is a linear combination of v1, v2, ... , vk-1. So set S can not be a linear independent set, which is a contradiction. So our assumption that T spans V must be false. That is T does not span V.
 
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